Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $q \neq 0$. $r = \dfrac{10(4q - 9)}{6q} \div \dfrac{16q - 36}{8} $
Dividing by an expression is the same as multiplying by its inverse. $r = \dfrac{10(4q - 9)}{6q} \times \dfrac{8}{16q - 36} $ When multiplying fractions, we multiply the numerators and the denominators. $r = \dfrac{ 10(4q - 9) \times 8 } { 6q \times (16q - 36) } $ $ r = \dfrac {8 \times 10(4q - 9)} {6q \times 4(4q - 9)} $ $ r = \dfrac{80(4q - 9)}{24q(4q - 9)} $ We can cancel the $4q - 9$ so long as $4q - 9 \neq 0$ Therefore $q \neq \dfrac{9}{4}$ $r = \dfrac{80 \cancel{(4q - 9})}{24q \cancel{(4q - 9)}} = \dfrac{80}{24q} = \dfrac{10}{3q} $